G7 Government Yields

Initialize environment

library(caim)
caim::clr()

Setup preferences

# most important maturities
maturities_a = c(.25, 2, 5, 10)
# interesting maturities, level b
maturities_b = c(.0833, 1, 30)
# all other maturities will be level c

# relative weightings for each maturity level
# these will form a density function that will be normalised to sum to 1
#   , so don't worry about that here
MAT_A_WT = 1
MAT_B_WT = .75
MAT_C_WT = 0.15

# set of maturities to use for lambda values
lambda_maturities <- seq(0.5, 5, 0.5)

Load G7 yield info

g7_rates_info <- fread("../data/ratesinfo.csv")[
  country %in% c("US", "CA", "DE", "FR", "IT", "UK", "JP") & class=="GOVT"
  , .(country, class, maturity, id=bb_ticker)
]

g7_rates_data <- g7_rates_info[
  fread("../data/ratesdata.csv")
  , 
  , on=.(id), nomatch=NULL 
][
  , .(country
      , class
      , maturity
      , date=lubridate::as_date(date)
      , year=lubridate::year(lubridate::as_date(date))
      , month=lubridate::month(lubridate::as_date(date))
      , yield=PX_LAST
      )
][
  , .SD, key=.(country, class, maturity, date)
]


g7_curve_mats <- sort(unique(g7_rates_data$maturity))
g7_curve_columns <- paste0("yield_", g7_curve_mats)

g7_curves <- dcast(
  g7_rates_data, country + class + date + year + month ~ maturity, value.var="yield"  
)[
  , .SD
  , key=.(country, class, date)  
]
setnames(g7_curves, c(key(g7_curves), g7_curve_mats), c(key(g7_curves), g7_curve_columns))

g7_curves[]

Load G7 Money market data

g7_mm_info <- data.table(
  country=c("US", "CA", "DE", "FR", "IT", "UK", "JP", "DE", "FR", "IT")
  , curncy=c("USD", "CAD", "EUR", "EUR", "EUR", "GBP", "JPY", "DEM", "FRF", "ITL")
  , id=c("USD.FIX.1M", "CAD.FIX.1M", "EUR.FIX.1M", "EUR.FIX.1M", "EUR.FIX.1M"
         , "GBP.FIX.1M", "JPY.FIX.1M", "DEM.FIX.1M", "FRF.FIX.1M", "ITL.FIX.1M")
)
# g7_mm_info <- fread("../data/mminfo.csv")[
#   CURNCY %in% c("USD",  "EUR", "GBP", "JPY", "CAD")
#   , .(curncy=CURNCY, id=FIX_1M)
# ]
g7_mm_data <- g7_mm_info[
  fread("../data/mmdata.csv")
  , 
  , on=.(id), nomatch=NULL
][
  , .(country
      , curncy
      , date = lubridate::as_date(date)
      , year=lubridate::year(lubridate::as_date(date))
      , month=lubridate::month(lubridate::as_date(date))
      , maturity = 0.8333
      , yield = PX_LAST
      )
][
  , .SD , key=.(country, curncy, date)
]

# stitch together pre- and post-EUR convergence data for DE, FR, IT
eur_start_date <- min(g7_mm_data[country=="DE" & curncy=="EUR"]$date)
# delete local observations where EUR data is available
g7_mm_data <- g7_mm_data[!(curncy %in% c("DEM", "FRF", "ITL") & date >= eur_start_date)]
# relabel pre-EUR local observations as EUR
# may need to rethink this if we want to implement pre- and post-EUR currency data
# - maybe label everything as DEM FRF ITL?
g7_mm_data[curncy %in% c("DEM", "FRF", "ITL"), curncy := "EUR"] 

g7_mm_data[]

Create monthly data

g7_curves_m <- g7_curves[
  date %in% caim::month_end_dates(date)
]

g7_mm_data_m <- g7_mm_data[
  date %in% caim::month_end_dates(date)
]

Add money market data to short end of government curves

This is, admittedly, a big kludge, but should help when there are no government yields below 2-year maturity, like early Canada, etc. Also, this may be a good way to average 3-month bills and 1-month deposits as a proxy for cash when doing Nelson-Siegel analyses.

Assumption: We’ll do LIBOR - 1/8 as a crude proxy for LIBID

g7_curves_m[g7_mm_data_m, yield_0.0833 := i.yield - 0.125, on=.(country, year, month)][]

Ensure there are at least 3 yield points

# ensure there are at least 3 yield points
g7_curves_m <- g7_curves_m[
  g7_curves_m[ ,.(valid=(sum(!is.na(.SD)) >= 3)), by=.(country, class, date)]$valid
][
  # find first date that works for all countries
  date >= max(g7_curves_m[,.(min_date=min(date)), by=.(country, class)]$min_date)
]

Calculate Nelson Siegel coefficients

curve_data <- g7_curves_m
curve_mats <- g7_curve_mats
curve_columns <- g7_curve_columns

curve_coefs <- list()
curve_ids <- unique(curve_data[,.(country, class)])
for (c in 1:nrow(curve_ids)) {
  t_curve_data <- curve_data[country == curve_ids[c, country] & class == curve_ids[c, class]]
  t_curve_dates <- sort(unique(t_curve_data$date))
  t_curve_mats <- curve_mats #sort(unique(yield_info[curve_id==c]$maturity))
  t_curve_mat_names <- curve_columns #as.character(t_curve_mats)
  
  t_curve_mat_wts <- rep(MAT_C_WT, length(t_curve_mats))
  names(t_curve_mat_wts) <- t_curve_mat_names
  t_curve_mat_wts[names(t_curve_mat_wts) %in% maturities_a] <- MAT_A_WT
  t_curve_mat_wts[names(t_curve_mat_wts) %in% maturities_b] <- MAT_B_WT
  t_curve_mat_wts <- t_curve_mat_wts / sum(t_curve_mat_wts)
  
  
  t_res <- list()
  for (i in 1:length(lambda_maturities)) {
    mat <- lambda_maturities[i]
    lambda <- caim::ns_mat2lambda(mat)
    print(paste("calculating for curve:", c, curve_ids[c, country], curve_ids[c, class], "maturity:", mat, "lambda:", lambda))
    t_res[[i]] <- list(
      mat=mat
      , lambda=lambda
      , coefs=data.table(
        country=curve_ids[c, country]
        , class=curve_ids[c, class]
        , date=t_curve_data[,date]
        , t(apply(t_curve_data, 1, function(x) 
          caim::ns_yields2coefs(t_curve_mats
                                , as.numeric(x[t_curve_mat_names])
                                , lambda=lambda
                                , wts=t_curve_mat_wts)))
        , lambda_mat=mat
        )
      )
  }
  
  # calculate summaries
  # 5 year halflife assuming 260 business days in a year
  decay <- halflife2decay(5*260)
  lambda_summary <- data.table(t(sapply(t_res, function(x) 
    c(mat=x$mat, lambda=x$lambda, wss=mean(x$coefs$wss), wss_exp=mean_exp(x$coefs$wss, decay)))))

  # choose best data
  best_hist_fit <- which(lambda_summary$wss == min(lambda_summary$wss))
  best_exp_fit <- which(lambda_summary$wss_exp == min(lambda_summary$wss_exp))
  # take average of best fits, but tilt towards best_exp_fit
  best_ix <- floor(mean(c(best_hist_fit, best_exp_fit))+ifelse(best_exp_fit > best_hist_fit, 0.5, 0))
  best_hist_coefs <- t_res[[best_ix]]$coefs
  # best_hist_coefs$curve_id <- c

  #add data to curve_coefs
  curve_coefs[[length(curve_coefs)+1]] <- best_hist_coefs
}

[1] "calculating for curve: 1 CA GOVT maturity: 0.5 lambda: 3.58657038690834"
[1] "calculating for curve: 1 CA GOVT maturity: 1 lambda: 1.79328063484169"
[1] "calculating for curve: 1 CA GOVT maturity: 1.5 lambda: 1.19552075559691"
[1] "calculating for curve: 1 CA GOVT maturity: 2 lambda: 0.896641550467829"
[1] "calculating for curve: 1 CA GOVT maturity: 2.5 lambda: 0.717323931054624"
[1] "calculating for curve: 1 CA GOVT maturity: 3 lambda: 0.59776033052309"
[1] "calculating for curve: 1 CA GOVT maturity: 3.5 lambda: 0.512348203420652"
[1] "calculating for curve: 1 CA GOVT maturity: 4 lambda: 0.448335176139679"
[1] "calculating for curve: 1 CA GOVT maturity: 4.5 lambda: 0.398496343358628"
[1] "calculating for curve: 1 CA GOVT maturity: 5 lambda: 0.358681854090179"
[1] "calculating for curve: 2 DE GOVT maturity: 0.5 lambda: 3.58657038690834"
[1] "calculating for curve: 2 DE GOVT maturity: 1 lambda: 1.79328063484169"
[1] "calculating for curve: 2 DE GOVT maturity: 1.5 lambda: 1.19552075559691"
[1] "calculating for curve: 2 DE GOVT maturity: 2 lambda: 0.896641550467829"
[1] "calculating for curve: 2 DE GOVT maturity: 2.5 lambda: 0.717323931054624"
[1] "calculating for curve: 2 DE GOVT maturity: 3 lambda: 0.59776033052309"
[1] "calculating for curve: 2 DE GOVT maturity: 3.5 lambda: 0.512348203420652"
[1] "calculating for curve: 2 DE GOVT maturity: 4 lambda: 0.448335176139679"
[1] "calculating for curve: 2 DE GOVT maturity: 4.5 lambda: 0.398496343358628"
[1] "calculating for curve: 2 DE GOVT maturity: 5 lambda: 0.358681854090179"
[1] "calculating for curve: 3 FR GOVT maturity: 0.5 lambda: 3.58657038690834"
[1] "calculating for curve: 3 FR GOVT maturity: 1 lambda: 1.79328063484169"
[1] "calculating for curve: 3 FR GOVT maturity: 1.5 lambda: 1.19552075559691"
[1] "calculating for curve: 3 FR GOVT maturity: 2 lambda: 0.896641550467829"
[1] "calculating for curve: 3 FR GOVT maturity: 2.5 lambda: 0.717323931054624"
[1] "calculating for curve: 3 FR GOVT maturity: 3 lambda: 0.59776033052309"
[1] "calculating for curve: 3 FR GOVT maturity: 3.5 lambda: 0.512348203420652"
[1] "calculating for curve: 3 FR GOVT maturity: 4 lambda: 0.448335176139679"
[1] "calculating for curve: 3 FR GOVT maturity: 4.5 lambda: 0.398496343358628"
[1] "calculating for curve: 3 FR GOVT maturity: 5 lambda: 0.358681854090179"
[1] "calculating for curve: 4 IT GOVT maturity: 0.5 lambda: 3.58657038690834"
[1] "calculating for curve: 4 IT GOVT maturity: 1 lambda: 1.79328063484169"
[1] "calculating for curve: 4 IT GOVT maturity: 1.5 lambda: 1.19552075559691"
[1] "calculating for curve: 4 IT GOVT maturity: 2 lambda: 0.896641550467829"
[1] "calculating for curve: 4 IT GOVT maturity: 2.5 lambda: 0.717323931054624"
[1] "calculating for curve: 4 IT GOVT maturity: 3 lambda: 0.59776033052309"
[1] "calculating for curve: 4 IT GOVT maturity: 3.5 lambda: 0.512348203420652"
[1] "calculating for curve: 4 IT GOVT maturity: 4 lambda: 0.448335176139679"
[1] "calculating for curve: 4 IT GOVT maturity: 4.5 lambda: 0.398496343358628"
[1] "calculating for curve: 4 IT GOVT maturity: 5 lambda: 0.358681854090179"
[1] "calculating for curve: 5 JP GOVT maturity: 0.5 lambda: 3.58657038690834"
[1] "calculating for curve: 5 JP GOVT maturity: 1 lambda: 1.79328063484169"
[1] "calculating for curve: 5 JP GOVT maturity: 1.5 lambda: 1.19552075559691"
[1] "calculating for curve: 5 JP GOVT maturity: 2 lambda: 0.896641550467829"
[1] "calculating for curve: 5 JP GOVT maturity: 2.5 lambda: 0.717323931054624"
[1] "calculating for curve: 5 JP GOVT maturity: 3 lambda: 0.59776033052309"
[1] "calculating for curve: 5 JP GOVT maturity: 3.5 lambda: 0.512348203420652"
[1] "calculating for curve: 5 JP GOVT maturity: 4 lambda: 0.448335176139679"
[1] "calculating for curve: 5 JP GOVT maturity: 4.5 lambda: 0.398496343358628"
[1] "calculating for curve: 5 JP GOVT maturity: 5 lambda: 0.358681854090179"
[1] "calculating for curve: 6 UK GOVT maturity: 0.5 lambda: 3.58657038690834"
[1] "calculating for curve: 6 UK GOVT maturity: 1 lambda: 1.79328063484169"
[1] "calculating for curve: 6 UK GOVT maturity: 1.5 lambda: 1.19552075559691"
[1] "calculating for curve: 6 UK GOVT maturity: 2 lambda: 0.896641550467829"
[1] "calculating for curve: 6 UK GOVT maturity: 2.5 lambda: 0.717323931054624"
[1] "calculating for curve: 6 UK GOVT maturity: 3 lambda: 0.59776033052309"
[1] "calculating for curve: 6 UK GOVT maturity: 3.5 lambda: 0.512348203420652"
[1] "calculating for curve: 6 UK GOVT maturity: 4 lambda: 0.448335176139679"
[1] "calculating for curve: 6 UK GOVT maturity: 4.5 lambda: 0.398496343358628"
[1] "calculating for curve: 6 UK GOVT maturity: 5 lambda: 0.358681854090179"
[1] "calculating for curve: 7 US GOVT maturity: 0.5 lambda: 3.58657038690834"
[1] "calculating for curve: 7 US GOVT maturity: 1 lambda: 1.79328063484169"
[1] "calculating for curve: 7 US GOVT maturity: 1.5 lambda: 1.19552075559691"
[1] "calculating for curve: 7 US GOVT maturity: 2 lambda: 0.896641550467829"
[1] "calculating for curve: 7 US GOVT maturity: 2.5 lambda: 0.717323931054624"
[1] "calculating for curve: 7 US GOVT maturity: 3 lambda: 0.59776033052309"
[1] "calculating for curve: 7 US GOVT maturity: 3.5 lambda: 0.512348203420652"
[1] "calculating for curve: 7 US GOVT maturity: 4 lambda: 0.448335176139679"
[1] "calculating for curve: 7 US GOVT maturity: 4.5 lambda: 0.398496343358628"
[1] "calculating for curve: 7 US GOVT maturity: 5 lambda: 0.358681854090179"

ns_coefs <- rbindlist(curve_coefs)[
  , .(ns_beta0 = beta0
      , ns_beta1 = beta1
      , ns_beta2 = beta2
      , ns_lambda = lambda
      , ns_lambda_mat = lambda_mat
      , ns_wss = wss
      )
  , key=.(country, class, date)
]

g7_curves_m <- g7_curves_m[ns_coefs]

rm(list=setdiff(ls(), c("g7_curves", "g7_curves_m", "g7_rates_data", "g7_rates_info", "g7_curve_columns", "g7_curve_mats")))

Yield calculations

long_data <- melt(
  g7_curves_m  
  , c("country", "class", "date", "ns_beta0", "ns_beta1", "ns_beta2", "ns_lambda")
  ,  patterns("yield_")
  , value.name="yield_now"
)[
  , maturity := as.numeric(stringr::str_remove(variable, "yield_"))
][
  , .(country, class, date, ns_beta0, ns_beta1, ns_beta2, ns_lambda, maturity, yield_now)
]

# fill in missing yields with interpolated data
# table with !is.na(yield)
yields_valid <- long_data[!is.na(yield_now)][, c("mat", "yld") := .(maturity, yield_now)]
# table mapping to <=
yields_lo <- yields_valid[long_data, .(country, date, maturity, m_lo0=mat, y_lo0=yld), on=.(country, date, maturity), roll=T]
# table mapping to >=
yields_hi <- yields_valid[long_data, .(country, date, maturity, m_hi0=mat, y_hi0=yld), on=.(country, date, maturity), roll=-Inf]

yields_lin <- yields_hi[
  yields_lo
  , .(
    country
    , date
    , maturity
    , yield_lin = ifelse(m_hi0 == m_lo0
                         , y_hi0
                         , y_lo0 + (maturity - m_lo0) * (y_hi0 - y_lo0) / (m_hi0 - m_lo0)
                         )
    , y_hi0
    , y_lo0
    , m_hi0
    , m_lo0
  )
  , on=.(country, date, maturity)
]

long_data <- long_data[yields_lin, on=.(country, date, maturity)]

long_data <- long_data[
  , c("num_coups", "mod_dur") := .(
    ifelse(maturity > 0.25, ifelse(country=="US", 2, 1), 0)
    , maturity
  )  
][
  maturity > 0.25
  , mod_dur := round(caim::modified_duration(yield_lin / 100, yield_lin / 100, maturity, num_coups), 6)
][
  , .(
    ns_beta0
    , ns_beta1
    , ns_beta2
    , ns_lambda
    , num_coups
    , mod_dur #= round(caim::modified_duration(yield_now / 100, yield_now / 100, maturity, 1), 6)
    , yield_now
    , yield_lin
    , y_hi0
    , y_lo0
    , m_hi0
    , m_lo0
    )
  , key=.(country, class, maturity, date)
][
  , yield_prev := shift(yield_lin), by = .(country, class, maturity)
][
  , c("y_hi1", "y_lo1", "m_hi1", "m_lo1") := .(
    yield_lin
    , shift(yield_lin)
    , maturity
    , shift(maturity)
  )
  , by = .(country, class, date)
][
  , yield_sell_lin := (
    y_lo1 + ((m_hi1 - 1/12) - m_lo1) * (y_hi1 - y_lo1) / (m_hi1 - m_lo1)
  )
][
  , yield_buy_ns := round(caim::ns_coefs2yields(
    mats = maturity
    , beta0 = shift(ns_beta0)
    , beta1 = shift(ns_beta1)
    , beta2 = shift(ns_beta2)
    , lambda = shift(ns_lambda)
  )$y, 4)
  , by = .(country, class, maturity)
][
  , yield_sell_ns := round(caim::ns_coefs2yields(
    mats = maturity - 1/2
    , beta0 = ns_beta0
    , beta1 = ns_beta1
    , beta2 = ns_beta2
    , lambda = ns_lambda
  )$y, 4)
][
  , coup_inc_lin := round(yield_prev / 1200, 6)
][
  , shift_inc := ifelse(maturity > 0.25, round((yield_lin - yield_prev) / 100 * -mod_dur, 6), 0)
][
  , tot_ret_shift := coup_inc_lin + shift_inc
][
  , dur_inc_lin := ifelse(
    maturity > 0.25
    , round(caim::bond_price(yield_sell_lin / 100, yield_prev / 100, maturity - 1/12, 1, 1) - 1, 6)
    , 0
    )
][
  , tot_ret_lin := coup_inc_lin + dur_inc_lin
][
  , coup_inc_ns := round(yield_buy_ns / 1200, 6)  
][
 , price_inc_ns := ifelse(
   maturity > 0.25
   , round(caim::bond_price(yield_sell_ns / 100, yield_buy_ns / 100, maturity - 1/12, 1, 1) - 1, 6)
   , 0
   )
][
  , tot_ret_ns := coup_inc_ns + price_inc_ns
]

long_data[country=="US"]

wide_data <- dcast(
  long_data
  ,country + class + date + ns_beta0 + ns_beta1 + ns_beta2 + ns_lambda ~ maturity
  , value.var = names(long_data)[!(names(long_data) %in% c("country", "class", "date", "ns_beta0", "ns_beta1", "ns_beta2", "ns_lambda", "maturity"))]

  # , value.var = c("mod_dur", "yield_now", "yield_prev", "yield_buy_ns", "yield_sell_ns"
  #                 , "coup_inc_lin", "dur_inc_lin", "coup_inc_ns", "price_inc_ns", "tot_ret_ns"
  #                 )
  )

wide_data[]

g7_return_data <- dcast(
  long_data[maturity %in% c(0.0833, 1, 2, 3, 5, 7, 10)]
  , country + class + date ~ maturity
  , value.var = c("coup_inc_lin", "tot_ret_shift", "tot_ret_lin", "tot_ret_ns")
)

g7_return_data[]

Get asset data

g7_assets <- data.table(
  id=c("US.GOVT.13", "US.GOVT.15", "US.GOVT.110"
       , "CA.GOVT.13", "CA.GOVT.15", "CA.GOVT.110"
       , "DE.GOVT.13", "DE.GOVT.15", "DE.GOVT.110"
       , "FR.GOVT.13", "FR.GOVT.15", "FR.GOVT.110"
       , "IT.GOVT.13", "IT.GOVT.15", "IT.GOVT.110"
       , "UK.GOVT.13", "UK.GOVT.15", "UK.GOVT.110"
       , "JP.GOVT.13", "JP.GOVT.15", "JP.GOVT.110"
       )
  , country=c(rep("US", 3), rep("CA", 3), rep("DE", 3), rep("FR", 3), rep("IT", 3)
              , rep("UK", 3), rep("JP", 3))
  , maturity=rep(c(13, 15, 110), 7)
  , key=c("country", "maturity")
)
g7_asset_info <- fread("../data/assetinfo.csv")[suggname %in% g7_assets$id]

g7_asset_info[]

g7_asset_data <- g7_assets[
  fread("../data/assetdata.csv")
  , .(
    country
    , maturity
    , id
    , date = lubridate::as_date(date)
    , year = lubridate::year(date)
    , month = lubridate::month(date)
    , index_nav=PX_LAST
    )
  , on=.(id), nomatch=NULL
][
  date %in% caim::month_end_dates(date)
  , .SD
  , key=.(country, maturity, date)
][
  , index_ret := round(index_nav / shift(index_nav) - 1, 6)
  , by=.(country, maturity)
]

|--------------------------------------------------|
|==================================================|

g7_asset_data[]

Model Setup

g7_models <- list(
  list(name="x_g13_coup_bullet", maturity=13, features=c("coup_inc_lin_2"), unity_coefs=T)
  , list(name="x_g13_coup_bullet", maturity=13, features=c("tot_ret_shift_2"), unity_coefs=T)
  , list(name="x_g13_lin_bullet", maturity=13, features=c("tot_ret_lin_2"), unity_coefs=T)
  , list(name="x_g13_ns_bullet", maturity=13, features=c("tot_ret_ns_2"), unity_coefs=T)
  , list(name="x_g13_coup_n", maturity=13, features=c("coup_inc_lin_1", "coup_inc_lin_2", "coup_inc_lin_3"), unity_coefs=T)
  , list(name="x_g13_shift_n", maturity=13, features=c("tot_ret_shift_1", "tot_ret_shift_2", "tot_ret_shift_3"), unity_coefs=T)
  , list(name="x_g13_lin_n", maturity=13, features=c("tot_ret_lin_1", "tot_ret_lin_2", "tot_ret_lin_3"), unity_coefs=T)
  , list(name="x_g13_ns_n", maturity=13, features=c("tot_ret_ns_1", "tot_ret_ns_2", "tot_ret_ns_3"), unity_coefs=T)
  , list(name="x_g13_coup_ladder", maturity=13, features=c("coup_inc_lin_1", "coup_inc_lin_2", "coup_inc_lin_3"), unity_coefs=F)
  , list(name="x_g13_shift_ladder", maturity=13, features=c("tot_ret_shift_1", "tot_ret_shift_2", "tot_ret_shift_3"), unity_coefs=F)
  , list(name="x_g13_lin_ladder", maturity=13, features=c("tot_ret_lin_1", "tot_ret_lin_2", "tot_ret_lin_3"), unity_coefs=F)
  , list(name="x_g13_ns_ladder", maturity=13, features=c("tot_ret_ns_1", "tot_ret_ns_2", "tot_ret_ns_3"), unity_coefs=F)

  , list(name="x_g15_coup_bullet", maturity=15, features=c("coup_inc_lin_3"), unity_coefs=T)
  , list(name="x_g15_coup_bullet", maturity=15, features=c("tot_ret_shift_3"), unity_coefs=T)
  , list(name="x_g15_lin_bullet", maturity=15, features=c("tot_ret_lin_3"), unity_coefs=T)
  , list(name="x_g15_ns_bullet", maturity=15, features=c("tot_ret_ns_3"), unity_coefs=T)
  , list(name="x_g15_coup_n", maturity=15, features=c("coup_inc_lin_1", "coup_inc_lin_2", "coup_inc_lin_3", "coup_inc_lin_5"), unity_coefs=T)
  , list(name="x_g15_shift_n", maturity=15, features=c("tot_ret_shift_1", "tot_ret_shift_2", "tot_ret_shift_3", "tot_ret_shift_5"), unity_coefs=T)
  , list(name="x_g15_lin_n", maturity=15, features=c("tot_ret_lin_1", "tot_ret_lin_2", "tot_ret_lin_3", "tot_ret_lin_5"), unity_coefs=T)
  , list(name="x_g15_ns_n", maturity=15, features=c("tot_ret_ns_1", "tot_ret_ns_2", "tot_ret_ns_3", "tot_ret_ns_5"), unity_coefs=T)
  , list(name="x_g15_coup_ladder", maturity=15, features=c("coup_inc_lin_1", "coup_inc_lin_2", "coup_inc_lin_3", "coup_inc_lin_5"), unity_coefs=F)
  , list(name="x_g15_shift_ladder", maturity=15, features=c("tot_ret_shift_1", "tot_ret_shift_2", "tot_ret_shift_3", "tot_ret_shift_5"), unity_coefs=F)
  , list(name="x_g15_lin_ladder", maturity=15, features=c("tot_ret_lin_1", "tot_ret_lin_2", "tot_ret_lin_3", "tot_ret_lin_5"), unity_coefs=F)
  , list(name="x_g15_ns_ladder", maturity=15, features=c("tot_ret_ns_1", "tot_ret_ns_2", "tot_ret_ns_3", "tot_ret_ns_5"), unity_coefs=F)
  
  , list(name="x_g110_coup_bullet", maturity=110, features=c("coup_inc_lin_5"), unity_coefs=T)
  , list(name="x_g110_coup_bullet", maturity=110, features=c("tot_ret_shift_5"), unity_coefs=T)
  , list(name="x_g110_lin_bullet", maturity=110, features=c("tot_ret_lin_5"), unity_coefs=T)
  , list(name="x_g110_ns_bullet", maturity=110, features=c("tot_ret_ns_5"), unity_coefs=T)
  , list(name="x_g110_coup_n", maturity=110, features=c("coup_inc_lin_1", "coup_inc_lin_2", "coup_inc_lin_3", "coup_inc_lin_5", "coup_inc_lin_7", "coup_inc_lin_10"), unity_coefs=T)
  , list(name="x_g110_shift_n", maturity=110, features=c("tot_ret_shift_1", "tot_ret_shift_2", "tot_ret_shift_3", "tot_ret_shift_5", "tot_ret_shift_7", "tot_ret_shift_10"), unity_coefs=T)
  , list(name="x_g110_lin_n", maturity=110, features=c("tot_ret_lin_1", "tot_ret_lin_2", "tot_ret_lin_3", "tot_ret_lin_5", "tot_ret_lin_7", "tot_ret_lin_10"), unity_coefs=T)
  , list(name="x_g110_ns_n", maturity=110, features=c("tot_ret_ns_1", "tot_ret_ns_2", "tot_ret_ns_3", "tot_ret_ns_5", "tot_ret_ns_7", "tot_ret_ns_10"), unity_coefs=T)
  , list(name="x_g110_coup_ladder", maturity=110, features=c("coup_inc_lin_1", "coup_inc_lin_2", "coup_inc_lin_3", "coup_inc_lin_5", "coup_inc_lin_7", "coup_inc_lin_10"), unity_coefs=F)
  , list(name="x_g110_shift_ladder", maturity=110, features=c("tot_ret_shift_1", "tot_ret_shift_2", "tot_ret_shift_3", "tot_ret_shift_5", "tot_ret_shift_7", "tot_ret_shift_10"), unity_coefs=F)
  , list(name="x_g110_lin_ladder", maturity=110, features=c("tot_ret_lin_1", "tot_ret_lin_2", "tot_ret_lin_3", "tot_ret_lin_5", "tot_ret_lin_7", "tot_ret_lin_10"), unity_coefs=F)
  , list(name="x_g110_ns_ladder", maturity=110, features=c("tot_ret_ns_1", "tot_ret_ns_2", "tot_ret_ns_3", "tot_ret_ns_5", "tot_ret_ns_7", "tot_ret_ns_10"), unity_coefs=F)
)

g7_features <- sort(unique(rbindlist(lapply(g7_models, function(x) data.table(feature=x$features))))$feature)

# g13_models[["x_g13_coup_bullet"]] <- list(features=c("coup_inc_lin_2"), unity_coefs=T)
# g13_models[["x_g13_shift_bullet"]] <- list(features=c("tot_ret_shift_2"), unity_coefs=T)
# g13_models[["x_g13_lin_bullet"]] <- list(features=c("tot_ret_lin_2"), unity_coefs=T)
# g13_models[["x_g13_ns_bullet"]] <- list(features=c("tot_ret_ns_2"), unity_coefs=T)
# 
# g13_models[["x_g13_coup_n"]] <- list(features=c("coup_inc_lin_1", "coup_inc_lin_2", "coup_inc_lin_3"), unity_coefs=T)
# g13_models[["x_g13_shift_n"]] <- list(features=c("tot_ret_shift_1", "tot_ret_shift_2", "tot_ret_shift_3"), unity_coefs=T)
# g13_models[["x_g13_lin_n"]] <- list(features=c("tot_ret_lin_1", "tot_ret_lin_2", "tot_ret_lin_3"), unity_coefs=T)
# g13_models[["x_g13_ns_n"]] <- list(features=c("tot_ret_ns_1", "tot_ret_ns_2", "tot_ret_ns_3"), unity_coefs=T)
# 
# g13_models[["x_g13_coup_ladder"]] <- list(features=c("coup_inc_lin_1", "coup_inc_lin_2", "coup_inc_lin_3"), unity_coefs=F)
# g13_models[["x_g13_shift_ladder"]] <- list(features=c("tot_ret_shift_1", "tot_ret_shift_2", "tot_ret_shift_3"), unity_coefs=F)
# g13_models[["x_g13_lin_ladder"]] <- list(features=c("tot_ret_lin_1", "tot_ret_lin_2", "tot_ret_lin_3"), unity_coefs=F)
# g13_models[["x_g13_ns_ladder"]] <- list(features=c("tot_ret_ns_1", "tot_ret_ns_2", "tot_ret_ns_3"), unity_coefs=F)
# 

Model evaluation

fah[]

[[1]]
[[1]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.189468  0.000021  0.004588        NA        NA -0.000601  0.004556 -0.007186  0.015783 

[[1]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.949463  0.000002  0.001497        NA        NA -0.000299  0.001469 -0.003586  0.005089 

[[1]][[3]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.952774  0.000002  0.001274        NA        NA -0.000011  0.001276 -0.000134  0.004422 

[[1]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.864476 0.000007 0.002645       NA       NA 0.001754 0.001982 0.021253 0.006867 

[[1]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.191281  0.000021  0.004588        NA        NA -0.000612  0.004554 -0.007318  0.015777 

[[1]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.954642  0.000002  0.001263        NA        NA -0.000314  0.001225 -0.003759  0.004244 

[[1]][[7]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.959402  0.000001  0.001076        NA        NA -0.000028  0.001078 -0.000333  0.003733 

[[1]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.882299 0.000006 0.002441       NA       NA 0.001642 0.001810 0.019877 0.006271 

[[1]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.189468  0.000022  0.004676        NA        NA -0.001084  0.004556 -0.012934  0.015783 

[[1]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.955921  0.000001  0.001100        NA        NA -0.000271  0.001067 -0.003244  0.003698 

[[1]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.959996  0.000001  0.001029        NA        NA -0.000164  0.001017 -0.001972  0.003524 

[[1]][[12]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.885104 0.000003 0.001797       NA       NA 0.000479 0.001735 0.005768 0.006010 


[[2]]
[[2]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.132916  0.000038  0.006193        NA        NA -0.000797  0.006151 -0.009526  0.021308 

[[2]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.922929  0.000007  0.002693        NA        NA -0.000323  0.002678 -0.003868  0.009277 

[[2]][[3]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.927751 0.000006 0.002497       NA       NA 0.000051 0.002501 0.000612 0.008664 

[[2]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.921147 0.000012 0.003426       NA       NA 0.002446 0.002403 0.029744 0.008324 

[[2]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.131729  0.000039  0.006210        NA        NA -0.000862  0.006160 -0.010300  0.021339 

[[2]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.956016  0.000002  0.001522        NA        NA -0.000441  0.001459 -0.005274  0.005054 

[[2]][[7]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.958895  0.000002  0.001359        NA        NA -0.000114  0.001356 -0.001371  0.004698 

[[2]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.924947 0.000007 0.002628       NA       NA 0.001860 0.001859 0.022553 0.006439 

[[2]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.130520  0.000039  0.006269        NA        NA -0.001225  0.006158 -0.014602  0.021333 

[[2]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.951667  0.000002  0.001506        NA        NA -0.000364  0.001463 -0.004358  0.005069 

[[2]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.954714  0.000002  0.001428        NA        NA -0.000216  0.001414 -0.002591  0.004897 

[[2]][[12]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.928875 0.000003 0.001796       NA       NA 0.000404 0.001753 0.004856 0.006072 


[[3]]
[[3]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.077622  0.000079  0.008879        NA        NA -0.000988  0.008838 -0.011795  0.030617 

[[3]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.953403  0.000011  0.003389        NA        NA -0.000195  0.003389 -0.002339  0.011739 

[[3]][[3]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.952457 0.000011 0.003261       NA       NA 0.000252 0.003257 0.003024 0.011282 

[[3]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.938262 0.000025 0.004988       NA       NA 0.003142 0.003880 0.038363 0.013441 

[[3]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.077686  0.000079  0.008898        NA        NA -0.001110  0.008843 -0.013239  0.030632 

[[3]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.969819  0.000003  0.001769        NA        NA -0.000377  0.001731 -0.004509  0.005997 

[[3]][[7]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.970536 0.000003 0.001648       NA       NA 0.000007 0.001651 0.000084 0.005718 

[[3]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.956928 0.000009 0.002918       NA       NA 0.002102 0.002028 0.025514 0.007025 

[[3]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.076562  0.000080  0.008955        NA        NA -0.001505  0.008842 -0.017910  0.030629 

[[3]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.968438  0.000003  0.001808        NA        NA -0.000465  0.001750 -0.005568  0.006063 

[[3]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.969906  0.000003  0.001731        NA        NA -0.000305  0.001707 -0.003650  0.005913 

[[3]][[12]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.946728 0.000005 0.002198       NA       NA 0.000518 0.002139 0.006240 0.007411 


[[4]]
[[4]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.258641  0.000012  0.003420        NA        NA -0.000629  0.003368 -0.007519  0.011666 

[[4]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.955748  0.000001  0.000982        NA        NA -0.000188  0.000965 -0.002255  0.003344 

[[4]][[3]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.961878  0.000001  0.000827        NA        NA -0.000006  0.000829 -0.000069  0.002871 

[[4]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.852544 0.000005 0.002148       NA       NA 0.001423 0.001611 0.017211 0.005581 

[[4]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.258655  0.000012  0.003420        NA        NA -0.000623  0.003368 -0.007445  0.011667 

[[4]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.957990  0.000001  0.000907        NA        NA -0.000184  0.000889 -0.002205  0.003081 

[[4]][[7]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.965088 0.000001 0.000762       NA       NA 0.000043 0.000762 0.000515 0.002639 

[[4]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.870840 0.000004 0.002044       NA       NA 0.001410 0.001482 0.017054 0.005135 

[[4]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.260315  0.000011  0.003376        NA        NA -0.000370  0.003361 -0.004427  0.011642 

[[4]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.963406  0.000001  0.000755        NA        NA -0.000127  0.000746 -0.001522  0.002584 

[[4]][[11]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.969115 0.000000 0.000684       NA       NA 0.000023 0.000685 0.000278 0.002373 

[[4]][[12]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.834595 0.000003 0.001745       NA       NA 0.000732 0.001587 0.008815 0.005498 


[[5]]
[[5]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.142016  0.000025  0.005047        NA        NA -0.000930  0.004969 -0.011105  0.017214 

[[5]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.927627  0.000004  0.001889        NA        NA -0.000284  0.001871 -0.003402  0.006481 

[[5]][[3]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.934587 0.000003 0.001711       NA       NA 0.000063 0.001713 0.000757 0.005933 

[[5]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.926171 0.000009 0.003060       NA       NA 0.002514 0.001748 0.030588 0.006055 

[[5]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.141639  0.000026  0.005057        NA        NA -0.000963  0.004973 -0.011497  0.017226 

[[5]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.952708  0.000002  0.001249        NA        NA -0.000376  0.001193 -0.004508  0.004131 

[[5]][[7]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.955835  0.000001  0.001129        NA        NA -0.000081  0.001128 -0.000972  0.003908 

[[5]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.930018 0.000005 0.002338       NA       NA 0.001854 0.001427 0.022477 0.004945 

[[5]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.146361  0.000025  0.004977        NA        NA -0.000506  0.004959 -0.006060  0.017179 

[[5]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.959687  0.000001  0.001091        NA        NA -0.000149  0.001082 -0.001787  0.003748 

[[5]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.962197  0.000001  0.001042        NA        NA -0.000058  0.001042 -0.000701  0.003611 

[[5]][[12]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.928945 0.000002 0.001533       NA       NA 0.000464 0.001464 0.005585 0.005071 


[[6]]
[[6]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.067152  0.000059  0.007669        NA        NA -0.001308  0.007569 -0.015582  0.026219 

[[6]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.893502  0.000013  0.003645        NA        NA -0.000277  0.003641 -0.003318  0.012613 

[[6]][[3]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.892557 0.000013 0.003543       NA       NA 0.000224 0.003542 0.002696 0.012270 

[[6]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.945179 0.000023 0.004812       NA       NA 0.003863 0.002874 0.047351 0.009954 

[[6]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.064771  0.000059  0.007691        NA        NA -0.001385  0.007578 -0.016496  0.026252 

[[6]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.889161  0.000008  0.002823        NA        NA -0.000431  0.002795 -0.005163  0.009681 

[[6]][[7]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.888709  0.000008  0.002745        NA        NA -0.000040  0.002749 -0.000480  0.009524 

[[6]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.965868 0.000008 0.002799       NA       NA 0.002363 0.001501 0.028732 0.005201 

[[6]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.068139  0.000057  0.007564        NA        NA -0.000441  0.007563 -0.005278  0.026200 

[[6]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.896386  0.000007  0.002558        NA        NA -0.000042  0.002562 -0.000508  0.008875 

[[6]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.894292  0.000007  0.002574        NA        NA -0.000047  0.002578 -0.000560  0.008931 

[[6]][[12]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.967268 0.000002 0.001438       NA       NA 0.000237 0.001421 0.002845 0.004923 


[[7]]
[[7]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.231993  0.000012  0.003506        NA        NA -0.000636  0.003453 -0.007611  0.011962 

[[7]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.908805  0.000002  0.001402        NA        NA -0.000244  0.001383 -0.002925  0.004791 

[[7]][[3]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.917408 0.000001 0.001225       NA       NA 0.000029 0.001226 0.000348 0.004248 

[[7]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.816217 0.000005 0.002302       NA       NA 0.001470 0.001775 0.017782 0.006149 

[[7]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.233744  0.000012  0.003504        NA        NA -0.000646  0.003450 -0.007729  0.011951 

[[7]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.931390  0.000001  0.001180        NA        NA -0.000253  0.001154 -0.003030  0.003998 

[[7]][[7]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.936686  0.000001  0.001039        NA        NA -0.000023  0.001040 -0.000280  0.003603 

[[7]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.842242 0.000005 0.002173       NA       NA 0.001451 0.001621 0.017550 0.005614 

[[7]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.231993  0.000012  0.003478        NA        NA -0.000461  0.003453 -0.005519  0.011962 

[[7]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.935031  0.000001  0.001015        NA        NA -0.000166  0.001004 -0.001988  0.003476 

[[7]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.939444  0.000001  0.000975        NA        NA -0.000096  0.000972 -0.001157  0.003367 

[[7]][[12]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.805760 0.000003 0.001813       NA       NA 0.000509 0.001743 0.006128 0.006037 


[[8]]
[[8]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.119365  0.000028  0.005278        NA        NA -0.000941  0.005203 -0.011230  0.018022 

[[8]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.895457  0.000005  0.002204        NA        NA -0.000361  0.002178 -0.004322  0.007545 

[[8]][[3]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.900149  0.000004  0.002037        NA        NA -0.000011  0.002041 -0.000136  0.007069 

[[8]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.895486 0.000010 0.003118       NA       NA 0.002445 0.001939 0.029733 0.006718 

[[8]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.119190  0.000028  0.005286        NA        NA -0.000977  0.005204 -0.011662  0.018027 

[[8]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.943907  0.000002  0.001405        NA        NA -0.000451  0.001333 -0.005393  0.004616 

[[8]][[7]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.946785  0.000002  0.001282        NA        NA -0.000122  0.001279 -0.001467  0.004429 

[[8]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.905828 0.000006 0.002515       NA       NA 0.001856 0.001700 0.022498 0.005890 

[[8]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.126021  0.000027  0.005209        NA        NA -0.000563  0.005187 -0.006730  0.017968 

[[8]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.934292  0.000002  0.001449        NA        NA -0.000205  0.001437 -0.002455  0.004978 

[[8]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.938960  0.000002  0.001410        NA        NA -0.000138  0.001405 -0.001653  0.004868 

[[8]][[12]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.911178 0.000003 0.001769       NA       NA 0.000085 0.001770 0.001025 0.006131 


[[9]]
[[9]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.053242  0.000068  0.008271        NA        NA -0.001380  0.008169 -0.016441  0.028297 

[[9]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.923597  0.000010  0.003183        NA        NA -0.000455  0.003156 -0.005447  0.010932 

[[9]][[3]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.923503 0.000009 0.003050       NA       NA 0.000169 0.003050 0.002031 0.010566 

[[9]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.906526 0.000024 0.004873       NA       NA 0.003659 0.003224 0.044803 0.011167 

[[9]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.051208  0.000069  0.008297        NA        NA -0.001484  0.008177 -0.017661  0.028325 

[[9]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.961551  0.000003  0.001772        NA        NA -0.000643  0.001654 -0.007690  0.005729 

[[9]][[7]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.963081  0.000003  0.001619        NA        NA -0.000170  0.001612 -0.002041  0.005585 

[[9]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.943097 0.000009 0.003032       NA       NA 0.002277 0.002005 0.027672 0.006944 

[[9]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.059610  0.000067  0.008173        NA        NA -0.000776  0.008150 -0.009268  0.028233 

[[9]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.958811  0.000003  0.001712        NA        NA -0.000118  0.001710 -0.001416  0.005925 

[[9]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.961451  0.000003  0.001657        NA        NA -0.000185  0.001650 -0.002220  0.005715 

[[9]][[12]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.946075  0.000004  0.002010        NA        NA -0.000466  0.001958 -0.005576  0.006784 


[[10]]
[[10]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.191719  0.000042  0.006480        NA        NA -0.000938  0.006423 -0.011202  0.022251 

[[10]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.947218  0.000004  0.001957        NA        NA -0.000265  0.001942 -0.003178  0.006729 

[[10]][[3]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.950546 0.000003 0.001720       NA       NA 0.000112 0.001720 0.001340 0.005957 

[[10]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.881665 0.000012 0.003459       NA       NA 0.002394 0.002500 0.029113 0.008661 

[[10]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.188908  0.000042  0.006493        NA        NA -0.000952  0.006433 -0.011363  0.022286 

[[10]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.964150  0.000002  0.001507        NA        NA -0.000299  0.001479 -0.003581  0.005124 

[[10]][[7]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.966702 0.000002 0.001323       NA       NA 0.000033 0.001324 0.000402 0.004588 

[[10]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.899475 0.000010 0.003197       NA       NA 0.002254 0.002272 0.027383 0.007870 

[[10]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.192101  0.000042  0.006469        NA        NA -0.000832  0.006426 -0.009943  0.022262 

[[10]][[10]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.950044 0.000003 0.001745       NA       NA 0.000220 0.001734 0.002637 0.006007 

[[10]][[11]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.950603 0.000003 0.001722       NA       NA 0.000172 0.001716 0.002066 0.005945 

[[10]][[12]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.909681  0.000005  0.002235        NA        NA -0.000420  0.002199 -0.005030  0.007616 


[[11]]
[[11]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.133502  0.000076  0.008694        NA        NA -0.001148  0.008632 -0.013690  0.029904 

[[11]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.953255  0.000008  0.002884        NA        NA -0.000201  0.002882 -0.002412  0.009985 

[[11]][[3]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.953062 0.000007 0.002675       NA       NA 0.000302 0.002663 0.003628 0.009224 

[[11]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.933523 0.000022 0.004645       NA       NA 0.003600 0.002940 0.044066 0.010183 

[[11]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.131402  0.000076  0.008714        NA        NA -0.001233  0.008641 -0.014698  0.029933 

[[11]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.966094  0.000003  0.001791        NA        NA -0.000386  0.001752 -0.004627  0.006068 

[[11]][[7]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.968467 0.000003 0.001635       NA       NA 0.000018 0.001638 0.000212 0.005675 

[[11]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.940604 0.000012 0.003531       NA       NA 0.002724 0.002250 0.033184 0.007794 

[[11]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.133502  0.000076  0.008730        NA        NA -0.001399  0.008632 -0.016664  0.029904 

[[11]][[10]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.961279 0.000005 0.002253       NA       NA 0.000529 0.002193 0.006366 0.007598 

[[11]][[11]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.963395 0.000004 0.002092       NA       NA 0.000292 0.002075 0.003509 0.007188 

[[11]][[12]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.945558  0.000006  0.002434        NA        NA -0.000732  0.002325 -0.008748  0.008054 


[[12]]
[[12]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.087204  0.000147  0.012121        NA        NA -0.001542  0.012043 -0.018345  0.041718 

[[12]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.953391  0.000020  0.004470        NA        NA -0.000114  0.004476 -0.001365  0.015506 

[[12]][[3]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.953613 0.000018 0.004281       NA       NA 0.000505 0.004259 0.006081 0.014753 

[[12]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.951694 0.000044 0.006623       NA       NA 0.004670 0.004703 0.057508 0.016293 

[[12]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.087566  0.000147  0.012139        NA        NA -0.001683  0.012042 -0.020015  0.041716 

[[12]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.966645  0.000006  0.002367        NA        NA -0.000458  0.002326 -0.005479  0.008058 

[[12]][[7]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.968507 0.000005 0.002224       NA       NA 0.000063 0.002227 0.000753 0.007714 

[[12]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.964060 0.000015 0.003857       NA       NA 0.003017 0.002407 0.036809 0.008337 

[[12]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.091780  0.000151  0.012295        NA        NA -0.002615  0.012035 -0.030932  0.041689 

[[12]][[10]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.969046 0.000007 0.002555       NA       NA 0.000706 0.002460 0.008510 0.008522 

[[12]][[11]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.970448 0.000006 0.002375       NA       NA 0.000310 0.002359 0.003723 0.008172 

[[12]][[12]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.964764  0.000007  0.002719        NA        NA -0.001116  0.002484 -0.013315  0.008605 


[[13]]
[[13]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.219293  0.000006  0.002468        NA        NA -0.000443  0.002432 -0.005304  0.008423 

[[13]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.870213  0.000001  0.001001        NA        NA -0.000224  0.000978 -0.002680  0.003387 

[[13]][[3]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.875400  0.000001  0.000943        NA        NA -0.000071  0.000942 -0.000854  0.003263 

[[13]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.775748 0.000003 0.001591       NA       NA 0.000796 0.001380 0.009594 0.004782 

[[13]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.222453  0.000006  0.002464        NA        NA -0.000441  0.002429 -0.005274  0.008414 

[[13]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.901902  0.000001  0.000874        NA        NA -0.000213  0.000849 -0.002557  0.002941 

[[13]][[7]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.909624  0.000001  0.000802        NA        NA -0.000054  0.000802 -0.000645  0.002778 

[[13]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.796224 0.000002 0.001566       NA       NA 0.000858 0.001312 0.010340 0.004545 

[[13]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.231389  0.000006  0.002422        NA        NA -0.000313  0.002406 -0.003747  0.008334 

[[13]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.901364  0.000001  0.000859        NA        NA -0.000213  0.000834 -0.002555  0.002888 

[[13]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.908864  0.000001  0.000803        NA        NA -0.000075  0.000801 -0.000896  0.002775 

[[13]][[12]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.790903 0.000002 0.001550       NA       NA 0.000918 0.001251 0.011068 0.004333 


[[14]]
[[14]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.161148  0.000017  0.004107        NA        NA -0.000782  0.004038 -0.009346  0.013989 

[[14]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.900221  0.000002  0.001460        NA        NA -0.000428  0.001398 -0.005129  0.004843 

[[14]][[3]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.908308  0.000002  0.001329        NA        NA -0.000174  0.001320 -0.002082  0.004574 

[[14]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.864549 0.000005 0.002266       NA       NA 0.001550 0.001655 0.018756 0.005735 

[[14]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.153073  0.000017  0.004120        NA        NA -0.000791  0.004050 -0.009453  0.014030 

[[14]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.909785  0.000002  0.001369        NA        NA -0.000452  0.001294 -0.005411  0.004484 

[[14]][[7]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.915251  0.000002  0.001285        NA        NA -0.000194  0.001272 -0.002323  0.004407 

[[14]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.862332 0.000004 0.002003       NA       NA 0.001233 0.001581 0.014892 0.005478 

[[14]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.153671  0.000016  0.004035        NA        NA -0.000465  0.004015 -0.005569  0.013908 

[[14]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.910073  0.000002  0.001333        NA        NA -0.000385  0.001278 -0.004610  0.004428 

[[14]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.915723  0.000002  0.001242        NA        NA -0.000134  0.001237 -0.001603  0.004284 

[[14]][[12]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.861512 0.000003 0.001779       NA       NA 0.000802 0.001591 0.009665 0.005512 


[[15]]
[[15]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.100856  0.000049  0.007030        NA        NA -0.001372  0.006907 -0.016336  0.023925 

[[15]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.868617  0.000008  0.002902        NA        NA -0.000697  0.002822 -0.008328  0.009777 

[[15]][[3]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.872371  0.000008  0.002756        NA        NA -0.000142  0.002757 -0.001709  0.009552 

[[15]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.880765 0.000015 0.003902       NA       NA 0.002829 0.002692 0.034481 0.009324 

[[15]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.096949  0.000050  0.007058        NA        NA -0.001413  0.006926 -0.016827  0.023993 

[[15]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.884530  0.000006  0.002541        NA        NA -0.000767  0.002427 -0.009170  0.008407 

[[15]][[7]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.887406  0.000006  0.002428        NA        NA -0.000347  0.002407 -0.004151  0.008339 

[[15]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.875978 0.000010 0.003096       NA       NA 0.001824 0.002507 0.022105 0.008683 

[[15]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.090787  0.000048  0.006918        NA        NA -0.000701  0.006894 -0.008382  0.023882 

[[15]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.887062  0.000006  0.002469        NA        NA -0.000623  0.002393 -0.007454  0.008289 

[[15]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.890474  0.000006  0.002369        NA        NA -0.000276  0.002356 -0.003309  0.008162 

[[15]][[12]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.871569 0.000007 0.002719       NA       NA 0.000952 0.002551 0.011488 0.008837 


[[16]]
[[16]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.233240  0.000016  0.004056        NA        NA -0.000604  0.004018 -0.007229  0.013918 

[[16]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.882893  0.000003  0.001767        NA        NA -0.000237  0.001754 -0.002837  0.006074 

[[16]][[3]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.876683  0.000003  0.001699        NA        NA -0.000119  0.001697 -0.001426  0.005880 

[[16]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.756171 0.000006 0.002464       NA       NA 0.000892 0.002301 0.010752 0.007970 

[[16]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.239139  0.000016  0.004041        NA        NA -0.000596  0.004004 -0.007125  0.013869 

[[16]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.908610  0.000002  0.001458        NA        NA -0.000208  0.001445 -0.002496  0.005007 

[[16]][[7]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.910988  0.000002  0.001381        NA        NA -0.000104  0.001379 -0.001245  0.004778 

[[16]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.782558 0.000005 0.002317       NA       NA 0.000864 0.002154 0.010415 0.007461 

[[16]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.237948  0.000016  0.004041        NA        NA -0.000568  0.004007 -0.006798  0.013881 

[[16]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.907193  0.000002  0.001454        NA        NA -0.000107  0.001453 -0.001279  0.005033 

[[16]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.910008  0.000002  0.001396        NA        NA -0.000073  0.001396 -0.000878  0.004837 

[[16]][[12]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.792889  0.000004  0.002088        NA        NA -0.000003  0.002092 -0.000032  0.007247 


[[17]]
[[17]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.122243  0.000035  0.005922        NA        NA -0.000868  0.005868 -0.010363  0.020327 

[[17]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.892193  0.000005  0.002264        NA        NA -0.000271  0.002252 -0.003243  0.007800 

[[17]][[3]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.893375 0.000005 0.002167       NA       NA 0.000014 0.002171 0.000172 0.007519 

[[17]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.825788 0.000009 0.003070       NA       NA 0.001427 0.002722 0.017260 0.009431 

[[17]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.123246  0.000035  0.005926        NA        NA -0.000913  0.005865 -0.010897  0.020318 

[[17]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.919275  0.000003  0.001817        NA        NA -0.000381  0.001779 -0.004558  0.006164 

[[17]][[7]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.923309  0.000003  0.001762        NA        NA -0.000225  0.001750 -0.002701  0.006064 

[[17]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.849578 0.000007 0.002633       NA       NA 0.001013 0.002435 0.012226 0.008434 

[[17]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.126121  0.000035  0.005888        NA        NA -0.000667  0.005860 -0.007972  0.020300 

[[17]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.919503  0.000003  0.001827        NA        NA -0.000093  0.001827 -0.001120  0.006330 

[[17]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.923453  0.000003  0.001759        NA        NA -0.000159  0.001754 -0.001906  0.006078 

[[17]][[12]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.866513  0.000005  0.002320        NA        NA -0.000399  0.002290 -0.004782  0.007931 


[[18]]
[[18]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.058906  0.000087  0.009321        NA        NA -0.001310  0.009244 -0.015607  0.032023 

[[18]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.925986  0.000009  0.002968        NA        NA -0.000344  0.002953 -0.004123  0.010231 

[[18]][[3]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.924766  0.000008  0.002870        NA        NA -0.000036  0.002875 -0.000436  0.009958 

[[18]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.853375 0.000021 0.004535       NA       NA 0.002015 0.004069 0.024451 0.014096 

[[18]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.057684  0.000087  0.009340        NA        NA -0.001402  0.009249 -0.016699  0.032040 

[[18]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.952357  0.000005  0.002145        NA        NA -0.000509  0.002087 -0.006087  0.007230 

[[18]][[7]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.955415  0.000004  0.002073        NA        NA -0.000296  0.002055 -0.003546  0.007119 

[[18]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.911731 0.000010 0.003084       NA       NA 0.001129 0.002875 0.013638 0.009959 

[[18]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.062435  0.000086  0.009261        NA        NA -0.000794  0.009242 -0.009492  0.032016 

[[18]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.947709  0.000005  0.002207        NA        NA -0.000148  0.002206 -0.001777  0.007642 

[[18]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.950554  0.000005  0.002177        NA        NA -0.000253  0.002166 -0.003027  0.007504 

[[18]][[12]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.922102  0.000008  0.002839        NA        NA -0.000755  0.002741 -0.009020  0.009496 


[[19]]
[[19]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.195397  0.000015  0.003898        NA        NA -0.000339  0.003889 -0.004060  0.013472 

[[19]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.978262  0.000001  0.001087        NA        NA -0.000238  0.001062 -0.002851  0.003679 

[[19]][[3]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.980917 0.000001 0.000787       NA       NA 0.000133 0.000777 0.001592 0.002690 

[[19]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.853075 0.000007 0.002641       NA       NA 0.001988 0.001741 0.024119 0.006030 

[[19]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.198572  0.000015  0.003893        NA        NA -0.000359  0.003883 -0.004295  0.013449 

[[19]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.978231  0.000001  0.001012        NA        NA -0.000252  0.000982 -0.003019  0.003401 

[[19]][[7]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.980417 0.000001 0.000746       NA       NA 0.000061 0.000745 0.000732 0.002580 

[[19]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.868232 0.000006 0.002498       NA       NA 0.001888 0.001639 0.022889 0.005679 

[[19]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.199217  0.000016  0.004031        NA        NA -0.001107  0.003883 -0.013202  0.013451 

[[19]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.981450  0.000000  0.000672        NA        NA -0.000130  0.000660 -0.001556  0.002287 

[[19]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.984750  0.000000  0.000579        NA        NA -0.000102  0.000571 -0.001218  0.001979 

[[19]][[12]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.880573  0.000002  0.001496        NA        NA -0.000009  0.001498 -0.000105  0.005189 


[[20]]
[[20]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.099492  0.000034  0.005815        NA        NA -0.000479  0.005804 -0.005734  0.020107 

[[20]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.969670  0.000004  0.002018        NA        NA -0.000291  0.002000 -0.003489  0.006929 

[[20]][[3]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.969321 0.000003 0.001738       NA       NA 0.000131 0.001736 0.001570 0.006014 

[[20]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.885919 0.000016 0.004006       NA       NA 0.003057 0.002593 0.037310 0.008984 

[[20]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.099370  0.000034  0.005821        NA        NA -0.000542  0.005805 -0.006479  0.020110 

[[20]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.982921  0.000001  0.001118        NA        NA -0.000354  0.001063 -0.004239  0.003681 

[[20]][[7]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.984458 0.000001 0.000850       NA       NA 0.000050 0.000850 0.000602 0.002943 

[[20]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.917367 0.000008 0.002912       NA       NA 0.002289 0.001803 0.027815 0.006245 

[[20]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.104212  0.000035  0.005934        NA        NA -0.001300  0.005799 -0.015491  0.020088 

[[20]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.982326  0.000001  0.000945        NA        NA -0.000168  0.000932 -0.002012  0.003227 

[[20]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.984489  0.000001  0.000852        NA        NA -0.000082  0.000850 -0.000983  0.002943 

[[20]][[12]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.930560 0.000003 0.001623       NA       NA 0.000214 0.001611 0.002576 0.005582 


[[21]]
[[21]][[1]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.053959  0.000070  0.008341        NA        NA -0.000534  0.008337 -0.006394  0.028881 

[[21]][[2]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.972311  0.000019  0.004366        NA        NA -0.000104  0.004372 -0.001246  0.015146 

[[21]][[3]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.969723 0.000017 0.004121       NA       NA 0.000574 0.004088 0.006905 0.014161 

[[21]][[4]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.902690 0.000040 0.006300       NA       NA 0.004049 0.004835 0.049679 0.016750 

[[21]][[5]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.051538  0.000070  0.008359        NA        NA -0.000681  0.008345 -0.008141  0.028908 

[[21]][[6]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.983529  0.000005  0.002295        NA        NA -0.000284  0.002281 -0.003405  0.007901 

[[21]][[7]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.983660 0.000004 0.002022       NA       NA 0.000185 0.002017 0.002217 0.006987 

[[21]][[8]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.946058 0.000013 0.003652       NA       NA 0.002632 0.002535 0.032047 0.008782 

[[21]][[9]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.058585  0.000071  0.008448        NA        NA -0.001443  0.008338 -0.017181  0.028883 

[[21]][[10]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.982650  0.000002  0.001260        NA        NA -0.000233  0.001240 -0.002797  0.004297 

[[21]][[11]]
       R2       MSE      RMSE      WMSE     WRMSE     alpha        te   alpha_y      te_y 
 0.984481  0.000001  0.001160        NA        NA -0.000155  0.001152 -0.001862  0.003990 

[[21]][[12]]
      R2      MSE     RMSE     WMSE    WRMSE    alpha       te  alpha_y     te_y 
0.952720 0.000004 0.001896       NA       NA 0.000332 0.001869 0.003987 0.006476 

Detour: constrained linear least squares = quadratic programming

evaluate <- function(model, per_in_year=12, wts=NULL) {
  y_hat <- model$X %*% model$coefs
  residuals <- y_hat - model$y
  return(
    c(
      R2 = cor(y_hat, model$y) ^ 2
      , MSE = mean(residuals^2)
      , RMSE = sqrt(mean(residuals^2))
      , WMSE = ifelse(!is.null(wts), t(model$wts) %*% residuals^2, NA)
      , WRMSE = ifelse(!is.null(wts), sqrt(t(model$wts) %*% residuals^2), NA)
      , alpha = mean(residuals)
      , te = sd(residuals)
      , alpha_y = (1 + mean(residuals)) ^ per_in_year - 1
      , te_y = sd(residuals) * sqrt(per_in_year)
    )
  )
}

t(sapply(names(g13_models), function(x) {
  model <- g13_models[[x]]
  y <- model_data[ , index_ret]
  X <- cbind(intercept=1, model_data[ , model$features, with=F])

  Aeq <- c(0, rep(1, ncol(X) - 1))
  beq <- 1
  lb <- c(caim::NEG_INF, rep(0, ncol(X) - 1))
  wts <- caim::d_exp(caim::halflife2decay(12), nrow(X))
  
  fixed_coefs <- NULL
  if (model$unity_coefs)
    fixed_coefs <- c(0, rep(1/length(model$features), length(model$features)))

  lmq <- caim::lm_quad(y, X, Aeq=Aeq, beq=beq, lb=lb, wts=wts, fixed_coefs=fixed_coefs)
  # coefs <- round(lmq$coefs, 6)
  # print(paste(x, coefs))
  
  print(lmq$coefs)
  return(round(evaluate(lmq), 6))
  # return(as.matrix(c(model=x, t(round(evaluate(lmq), 6)))))
}))

     intercept coup_inc_lin_2 
             0              1 
      intercept tot_ret_shift_2 
              0               1 
    intercept tot_ret_lin_2 
            0             1 
   intercept tot_ret_ns_2 
           0            1 
     intercept coup_inc_lin_1 coup_inc_lin_2 coup_inc_lin_3 
     0.0000000      0.3333333      0.3333333      0.3333333 
      intercept tot_ret_shift_1 tot_ret_shift_2 tot_ret_shift_3 
      0.0000000       0.3333333       0.3333333       0.3333333 
    intercept tot_ret_lin_1 tot_ret_lin_2 tot_ret_lin_3 
    0.0000000     0.3333333     0.3333333     0.3333333 
   intercept tot_ret_ns_1 tot_ret_ns_2 tot_ret_ns_3 
   0.0000000    0.3333333    0.3333333    0.3333333 
     intercept coup_inc_lin_1 coup_inc_lin_2 coup_inc_lin_3 
 -9.415735e-04  -1.629319e-15  -8.881784e-16   1.000000e+00 
      intercept tot_ret_shift_1 tot_ret_shift_2 tot_ret_shift_3 
   0.0001707531    0.3574075724    0.4676005452    0.1749918824 
    intercept tot_ret_lin_1 tot_ret_lin_2 tot_ret_lin_3 
-0.0001469486  0.2658323621  0.5647934936  0.1693741443 
    intercept  tot_ret_ns_1  tot_ret_ns_2  tot_ret_ns_3 
-1.598000e-03  5.809298e-01  5.716268e-18  4.190702e-01 
                         R2     MSE     RMSE WMSE WRMSE     alpha       te   alpha_y     te_y
x_g13_coup_bullet  0.195397 1.5e-05 0.003898   NA    NA -0.000339 0.003889 -0.004060 0.013472
x_g13_shift_bullet 0.978262 1.0e-06 0.001087   NA    NA -0.000238 0.001062 -0.002851 0.003679
x_g13_lin_bullet   0.980917 1.0e-06 0.000787   NA    NA  0.000133 0.000777  0.001592 0.002690
x_g13_ns_bullet    0.853075 7.0e-06 0.002641   NA    NA  0.001988 0.001741  0.024119 0.006030
x_g13_coup_n       0.198572 1.5e-05 0.003893   NA    NA -0.000359 0.003883 -0.004295 0.013449
x_g13_shift_n      0.978231 1.0e-06 0.001012   NA    NA -0.000252 0.000982 -0.003019 0.003401
x_g13_lin_n        0.980417 1.0e-06 0.000746   NA    NA  0.000061 0.000745  0.000732 0.002580
x_g13_ns_n         0.868232 6.0e-06 0.002498   NA    NA  0.001888 0.001639  0.022889 0.005679
x_g13_coup_ladder  0.199217 1.6e-05 0.004031   NA    NA -0.001107 0.003883 -0.013202 0.013451
x_g13_shift_ladder 0.981450 0.0e+00 0.000672   NA    NA -0.000130 0.000660 -0.001556 0.002287
x_g13_lin_ladder   0.984750 0.0e+00 0.000579   NA    NA -0.000102 0.000571 -0.001218 0.001979
x_g13_ns_ladder    0.880573 2.0e-06 0.001496   NA    NA -0.000009 0.001498 -0.000105 0.005189

# model <- g13_models[["x_g13_coup_bullet"]]
# 
# y <- model_data[ , index_ret]
# X <- cbind(intercept=1, g7_return_data[ , model$features, with=F])
# Aeq <- c(0, rep(1, ncol(X) - 1))
# beq <- 1
# lb <- c(caim::NEG_INF, rep(0, ncol(X) - 1))
# wts <- caim::d_exp(caim::halflife2decay(12), nrow(X))
# 
# lmq <- caim::lm_quad(y, X, Aeq=Aeq, beq=beq, lb=lb, wts=wts)
# coefs <- round(lmq$coefs, 6)
# print(coefs)

# set.seed(1967)
# N <- 100
# x1 <- runif(N)
# x2 <- runif(N)
# wts <- caim::d_exp(caim::halflife2decay(12), N)
# z <- seq(0, 1, length.out=N)
# y <- z * x1 + (1-z) * x2
# X <- cbind(x1, x2)
# Aeq <- rep(1, ncol(X))
# beq <- 1
# lb <- rep(0, ncol(X))
# 
# coefs <- round(caim::lm_quad(y, X, Aeq=Aeq, beq=beq, lb=lb), 6)
# print(coefs)
# 
# coefs <- round(caim::lm_quad(y, X, Aeq=Aeq, beq=beq, lb=lb, wts=wts), 6)
# print(coefs)